Editing Multimodal distribution (section)

===Bimodality coefficient===

Sarle's bimodality coefficient ''b'' is<ref name=Ellison1987>{{cite journal | last1 = Ellison | first1 = AM | year = 1987 | title = Effect of seed dimorphism on the density-dependent dynamics of experimental populations of ''Atriplex triangularis'' (Chenopodiaceae) | journal = American Journal of Botany | volume = 74 | issue = 8| pages = 1280–1288 | doi=10.2307/2444163| jstor = 2444163 }}</ref>

<math display="block"> \beta = \frac{ \gamma^2 + 1 }{ \kappa }  </math>

where ''γ'' is the [[skewness]] and ''κ'' is the [[kurtosis]]. The kurtosis is here defined to be the standardised fourth moment around the mean. The value of ''b'' lies between 0 and 1.<ref name=Pearson1916>{{cite journal | last1 = Pearson | first1 = K | year = 1916 | title = Mathematical contributions to the theory of evolution, XIX: Second supplement to a memoir on skew variation | journal = Philosophical Transactions of the Royal Society A | volume = 216 | issue = 538–548| pages = 429–457 | doi = 10.1098/rsta.1916.0009 | jstor = 91092 | bibcode =   1916RSPTA.216..429P| doi-access = free }}</ref> The logic behind this coefficient is that a bimodal distribution with light tails will have very low kurtosis, an asymmetric character, or both – all of which increase this coefficient.

The formula for a finite sample is<ref name=SASInst2012>SAS Institute Inc. (2012). SAS/STAT 12.1 user’s guide. Cary, NC: Author.</ref>

<math display="block"> b = \frac{ g^2 + 1 }{ k + \frac{ 3( n - 1 )^2 }{ ( n - 2 )( n - 3 ) } } </math>

where ''n'' is the number of items in the sample, ''g'' is the [[sample skewness]] and ''k'' is the sample [[excess kurtosis]].

The value of ''b'' for the [[uniform distribution (continuous)|uniform distribution]] is 5/9. This is also its value for the [[exponential distribution]]. Values greater than 5/9 may indicate a bimodal or multimodal distribution, though corresponding values can also result for heavily skewed unimodal distributions.<ref>{{cite journal | last1 = Pfister | first1 = R | last2 = Schwarz | first2 = KA | last3 = Janczyk | first3 = M. | last4 = Dale | first4 = R | last5 = Freeman | first5 = JB | year = 2013 | title = Good things peak in pairs: A note on the bimodality coefficient | journal =  Frontiers in Psychology| volume = 4 | pages = 700 | doi = 10.3389/fpsyg.2013.00700| pmid = 24109465 | pmc = 3791391 | doi-access = free }}</ref> The maximum value (1.0) is reached only by a [[Bernoulli distribution]] with only two distinct values or the sum of two different [[Dirac delta function]]s (a bi-delta distribution).

The distribution of this statistic is unknown. It is related to a statistic proposed earlier by Pearson – the difference between the kurtosis and the square of the skewness (''vide infra'').